Hodge-type integrals on moduli spaces of Admissible
Covers
Renzo Cavalieri
Department of Mathematics
University of Utah
155 South 1400 East, Room 233
Salt Lake City, UT 84112-0090
Email: renzo@math.utah.edu
Abstract
In this paper we study a natural class of intersection numbers on moduli
spaces of degree d Admissible Covers from genus g curves to P1 , using
techniques of localization. These intersection numbers involve tautological
and classes, and are in some sense analogous to Hodge Integrals on
moduli spaces of Stable Curves.
We compute explicitly these numbers for all genera in degrees 2 and 3 and
express the result in generating function form; we provide a conjecture for
the general degree d case.
AMS Classication 14N35
Keywords Hodge Integrals, Moduli Spaces, Localization, Admissible Covers
Introduction
Hodge integrals are a class of intersection numbers on moduli spaces of curves
involving the tautological classes i , which are the Chern classes of the Hodge
bundle E . In recent years Hodge integrals have shown a great amount of interconnections with Gromov-Witten theory and enumerative geometry.
The classical Hurwitz numbers, counting the numbers of ramied Covers of a
curve with an assigned set of ramication data, can be computed via Hodge
integrals. Simple Hurwitz numbers have been discussed in [4], [5] and [8];
progress towards double Hurwitz numbers has been made in [7].
Various spectacular computations of Hodge integrals were carried out in the
late nineties by Faber and Pandharipande ([6]). Their results have been used
to determine the multiple cover contributions in the GW invariants of P1 , thus
1
extending the well-known Aspinwall-Morrison formula in Gromov-Witten Theory.
Hodge integrals are also at the heart of the theory developed in [3], studying
the local Gromov-Witten theory of curves.
It is this last theory that brought our attention to a similar type of integrals.
We study moduli spaces of Admissible Covers, a natural compactication of
the Hurwitz scheme. It has been shown in [1] that these spaces are smooth
Deligne Mumford stacks. A class of natural intersection numbers on these
spaces, parallel (and we believe related) to the structure coecients of the
Topological Quantum Field Theory in [3], are obtained in the following way.
Consider the diagram of stacks:
f
,!
U
#
P1
Admg!d P ;( ;t ; ;tn )
1
where:
1
2
Admg!d P ;( ;t ; ;tn) denotes the space of (connected) genus g , degree
d , Admissible Covers, with ramication (1; t2 ; ; tn ), which we will
1
1
2
discuss at length in section 1.
we consider covers that have one arbitrary ramication point 1 ; all other
ramication is simple ( t stands for transposition);
U is the Universal family;
f is \morally" the Universal cover map (see page 7);
Now dene the class of integrals:
Id (g) :=
Z
ev1 (1) \ c2g+d,1 (R1 f (OP OP (,1)));
1
Admg!d P1 ;(
;tn )
1 ;t2 ;
1
where ev1 is evaluation at the rst marked point. (pag. 8)
It is an elementary dimension count to show that the only non vanishing integrals must have = (d), i.e. full ramication over the point 1 . For this
reason we drop the superscript .
We want to organize all of these integrals in generating function form:
1
X
Id(x) := 2g +Id (dg,) 1! x2g+d,1 :
g=0
2
These integrals can be approached with techniques of localization. We follow the
spirit, and also the notation, of Faber and Pandharipande( [6]), who pioneered
and developed the fundamental ideas of auxiliary localization integrals, and of
using dierent linearizations of line bundles as means to nd relations between
Hodge integrals.
We make the following conjecture.
Conjecture: for all d 1
Id
,
, 2 sin x2 d
d 2 sin , dx2 :
(x) = (,)d,1 1
The conjecture is trivially true for d = 1. In this paper we prove it for d = 2; 3.
Dierent strategies are required to prove these two results. In degree 2 we
exploit the fact that generic ramication is, in fact, full ramication. In degree
3 we prove the result by means of an auxiliary integral that we know to vanish;
we can obtain the auxiliary integral precisely because full ramication can be
thought of as a degeneration of simple ramication.
The strategy adopted in degree 3 should in principle work in higher degrees as
well. The problem in a direct computation is that the combinatorial complexity,
which is modest in the two cases we examine, grows dramatically fast.
As a corollary of these computations we obtain generating functions for another
interesting class of integrals:
Jd (g) :=
Z
c2g+2d,2 (R1 f (OP (,1) OP (,1))):
1
Admg!d P1 ;(t
1
;t2g+2d,2 )
1 ;t2 ;
In genus 0, we recover the Aspinwall-Morrison formula.
Acknowledgements
I am grateful rst and foremost to my advisor, Aaron Bertram, for his constant
support, motivation, and expert guidance. I also thank Y.P. Lee and Ravi Vakil
for carefully listening to my arguments and providing useful feedback.
3
1 Admissible Covers
Moduli Spaces of Admissible Covers are a \natural" compactication of the
Hurwitz scheme. The fundamental idea is that, in order to understand limit
covers, we allow the base curve to degenerate together with the cover. Branch
points are not allowed to \come together"; as two or more branch points tend to
collide, a new component of the base curve sprouts from the point of collision,
and the points transfer onto it. Similarly, upstairs the cover splits into a nodal
cover.
Now more formally: let (X; p1 ; ; pr ) be an r -pointed nodal curve of genus
g.
Denition 1 An admissible cover : E ,! X of degree d is a nite
morphism satisfying the following:
(1) E is a connected nodal curve.
(2) Every node of E maps to a node of X.
(3) The restriction of : E ,! X to X n (p1 ; ; pr ) is etale of constant
degree d.
(4) Over a node, locally in analytic coordinates, X, E and are described as
follows:
E : e1 e2 = a;
X : x1x2 = an;
: x1 = en1 ; x2 = en2 :
Moduli spaces of Admissible Covers were introduced originally by Harris and
Mumford in [9]. Intersection Theory on these spaces was for a long time extremely hard and mysterious, mostly because they are in general not normal,
even if the normalization is always smooth. Only recently in [1], Abramovich,
Corti and Vistoli exhibit this normalization as the stack of balanced stable maps
of degree 0 from twisted curves to the classifying stack BSd . This way they attain both the smoothness of the stack and a nice moduli-theoretic interpretation
of it.
We will abuse notation and refer to the Abramovich-Corti-Vistoli spaces as
Admissible Covers. We will be interested in admissible covers of P1 . In order
to estabilish notation, let us recall our basic denitions:
Denition 2 Fix d 1, and let 1; ; n be partitions of d. We denote by
Admg!d 0;( ; ;n )
1
4
the connected component of the stack of balanced stable maps of degree 0
from a genus 0, n-pointed twisted curve to BSd characterized by the following
conditions:
(1) the associated admissible cover (according to the construction in [1],
pag.3566) is a nodal curve of genus g.
(2) let x1 ; ; xn be the marks on the base curve; the ramication prole
over xi is required to be of type i .
We call this the stack of Admissible Covers of degree d and genus g of a genus
0 curve.
This
is either empty or a smooth stack of dimension n , 3 = 2g + 2d + n +
P
`(i ) , nd , 5, where `(i ) denotes the length of the partition i . It admits
two natural maps into moduli spaces of curves, as represented in the following
diagram:
Admg!d 0;( ; ;n ) ! M g
#
1
M 0;n
:
In particular, the vertical map has nite bers.
We also are interested in xing a parametrization of the base P1 . The objects
we parametrize are the same as above, but the equivalence relation is stricter:
we consider two covers E1 ! P1 , E2 ! P1 equivalent if there is an isomorphism
' : E1 ! E2 that makes the natural triangle commute. In other words, we are
not allowed to act on the base with an automorphism of P1 .
Denition 3 We denote by
Admg!d P ;( ; ;n)
1
1
the stack of admissible covers of degree d of (a parametrized) P1 by curves of
genus g, with n specied branch points having ramication prole 1 ; ; n .
We construct the space of parametrized admissible covers as the stack of balanced stable maps of degree d! from the category of genus 0, n -pointed twisted
curves to the stack quotient [P1 =Sd ], where Sd acts trivially on P1 . This is but
a slight variation to the A-C-V construction. Let us illustrate what happens
over a geometric point Spec( C ):
A map of degree d! from the twisted curve produces a map of degree 1 from the
coarse curve (and this is our desired parametrization of one special genus 0 twig
5
E
.
S -equivariant
d
degree d!
Trivial action
IP
1
d!
twisted curve
d!
degree d!
C
1
[P
I /Sd ]
1
1/d!
degree 1
C
IP
1
Spec C
Figure 1: the stack of admissible covers of a parametrized P1 .
on the base), a principal Sd bundle over the twisted curve and an Sd equivariant
map to P1 (this data characterizes the admissible cover). Two admissible covers
are equivalent if there is an automorphism of the twisted curve that makes them
commmute. In doing so, the degree 1 map to P1 has to be respected, so only
the non-parametrized twigs are free to be acted upon by automorphisms.
P
This is either empty or a smooth stack of dimension n = 2g +2d + n + `(i ) ,
nd , 2, admitting two natural morphisms
Admg!d P ;( ; ;n ) ! M g
1
#
1
P1 [n]
:
The map to M g just looks at the source curve forgetting the cover map. The
vertical morphism, taking values in the Fulton-Mac Pherson conguration space
of n points in P1 , looks instead at the target curve, and at the (ordered) branch
points.
The stacks of Admissible Covers of P1 admit a Universal family U , and a
Universal cover map . The cover map takes values in a stack X , that is a
family over the moduli space. The ber over a moduli point consists of a nodal,
genus 0 curve, with one special irreducible component. The Universal cover
map can be followed by a map " , that contracts all secondary twigs and takes
6
E
.
"special" twig
C
1
IP
1
Figure 2: schematic depiction of an admissible cover of a parametrized P1 .
values in Adm P1 . Finally the right projection lands us in P1 .
,!
X !" Admg!d P ;( ; ;n) P1 !
U
1
#
1
.
:
Admg!d P ;( ; ;n )
We call f the composition of the three horizontal maps.
1
P1
1
The Universal family can be itself interpreted as a moduli space of Admissible
Covers. If we think of admissible covers as of stable maps from a twisted
curve, then we obtain a Universal family by adding a mark to the twisted curve
and requiring trivial ramication over it. Let us denote with (1) the partition
(1; :::; 1) of d , representing an unramied point. Then,
U = Admg!d P ;( ; ;n;(1)) :
We can dene n tautological sections
i : Admg!d P ;( ; ;n ) ,! Admg!d P ;( ; ;n ;(1))
of the natural forgetful map. The image of the i -th section consists of covers
where a new rational component has sprouted from the i -th marked point. The
1
1
1
1
1
7
1
marked points (1) and i have transferred onto this twig. Over this twig we
nd `(i ) copies of P1 fully ramied over the attaching point and over the
marked point i .
E
E’
σi
C
µi
C’
(1)
µi
new twig
Figure 3: the tautological section i .
Finally we can dene the natural evaluation maps:
evi := f i : Admg!d P ;( ; ;n ) ,! P1 :
1
1
1.1 The boundary
The boundary of spaces of admissible covers can be described in terms of admissible cover spaces of possibly lower degree or genus. In the case of admissible
covers of a parametrized P1 , the boundary will involve also admissible covers
of an unparametrized genus 0 curve. In gure 1, for example, we can obtain
the depicted admissible cover by \gluing together" one admissible cover of a
parametrized P1 (the cover of the special twig) and three admissible covers
of an irreducible genus 0 curve. It would be very tempting to conclude that
the irreducible boundary components of an admissible cover space are actually
products of other admissible cover spaces; however we need to be very careful,
and consider the contribution to the stack structure given by automorphisms.
To illustrate this point let us carefully analyze the gluing map. For simplicity
8
of exposition, let's glue at a fully ramied point:
Admg !d P ;( ; ;n ;(d)) Admg !d 0;( ; ;n ;(d))
1
1
1
1
2
1
#
B ,! Admg +g !d P ;( ; ;n ; ; ;n )
1
2
2
1
1
1
1
2
We claim that the vertical map is an etale map of stacks of degree 1=d . Let us
look at a point [E ! X ] of B : we observe that it admits a unique preimage
([E1 ! P1 ]; [E2 ! X2 ]), and we count the automorphisms of the preimage
modulo automorphisms pulled-back from below. In local analytic coordinates
around the node, the cover is described as
Spec(C[e1 ; e2 ]=(e1 e2 , a))
#
Spec(C[x1 ; x2 ]=(x1 x2 , ad ));
by the local equations x1 = ed1 ; x2 = ed2 . Modding out by automorphisms of
the \glued" cover is equivalent to requiring the rst coordinate e1 to remain
untouched. It is then evident that what we have left are d distinct automorphisms, consisting in multiplying e2 by a dth root of unity. This establishes
our claim.
Now if we want to glue two branch points with ramication prole = (d1 ; :::; dk ),
with all the di 's distinct, the situation will be analogous. The gluing map
Admg !d P ;( ; ;n ;) Admg !d 0;( ; ;n ;)
1
1
1
2
1
#
B0 ,! Admg +g +`(),1!d P ;( ; ;n ; ; ;n )
is an etale map of stacks of degree 1=(d1 ::: dk ).
1
1
2
2
1
1
1
1
(1)
2
For the purposes of this paper, this is all we are concerned with. For the sake of
a more complete exposition, we briey describe what the situation for a general
partition is. Let
= ((1 )m ; : : : ; (k )mk ):
The gluing map (1) is an etale map of stacks of degree
1
Q
(2)
(i )mi mi ! :
There is a little bit of combinatorial subtlety to be dealt with to obtain this
last result. In order to be able to even dene a gluing map we must introduce
markings on the covers. Ionel develops the theory of these spaces in [10]; our
spaces are etale quotients of Ionel spaces. The gluing map is well dened on
the level of Ionel spaces, and it descends to (2).
1
9
1.2 Tautological Classes
We are interested in describing some \tautological" intersection classes on the
stack of admissible Covers of an unparametrized genus 0 curve: in particular
we want to endow our space with analogues of and classes. To do so, we
will simply pull-back these classes from the appropriate moduli spaces.
Recall the forgetful map:
The tautological class
Hodge bundle E .
Admg!d 0;( ; ;n ) !s M g :
i 2 Ai (M g )
1
is dened to be th i -th Chern class of the
Denition 4 The tautological class Adm
2 Ai (Admg!d 0;( ; ;n) ) is dened
i
to be the i -th Chern class of the pull-back of the Hodge bundle via the map s :
Adm
:= s (i ):
i
1
We will drop the superscript \Adm" and simply write i whenever there is no
risk of confusion.
Let us now look at another natural map:
Admg!d 0;( ; ;n ) !t M0;n :
1
The stack M0;n is the moduli space of twisted n-pointed curves of genus 0.
Let M0;n+1 ,!
M0;n be the universal family over this stack, ! ! M0;n+1
be the relative dualizing sheaf and i the i -th tautological section. Then i 2
A1 (M0;n) is dened to be the rst Chern class of i ( ! ).
Denition 5 The tautological class
Adm 2 A1 (Adm d
i
g!0;(1 ; ;n ) )
to be the pull-back of the analogous class via the map t :
Adm := t ( ):
i
i
Again, the superscript will be dropped unless needed for clarity.
We can also view classes in a more intrinsic fashion. Consider:
the space
Admg!d 0;( ; ;n ;(1))
1
where we have added a trivial ramication condition;
10
is dened
the forgetful map
(1) : Admg!d 0;( ; ;n ;(1)) ,! Admg!d 0;( ; ;n) ;
the i -th tautological section
1
1
i : Admg!d 0;( ; ;n ) ,! Admg!d 0;( ; ;n ;(1)) :
1
1
Lemma 1 The class , i 2 A1 (Admg!d 0;( ; ;n ) ) is the rst Chern class of
the normal bundle to the image of the section i .
1
proof:
Observe the following commutative diagram:
Admg!d 0;( ; ;n ;(1))
t~
,!
M0;n+1
~i "# (1)
2
i "# Admg!d 0;( ; ;n )
t
,!
M0;n
1
1
We know from [1], pag 3561, that the maps t and t~ are etale onto their image.
Further, the diagram is cartesian. Now our lemma follows from the analogous
statement on M0;n :
, iAdm = t(, i ) = c1 (t iNi ) = c1 (~i t~Ni ) = c1 (~i N~i ):
2 Localization
The main tool for evaluating our integrals is the Atiyah-Bott localization theorem ( [2]). Consider the 1 dimensional algebraic torus C , and recall that the
C -equivariant Chow ring of a point is a polynomial ring in one variable:
AC (fptg; C) = C[~]:
Let C act on a smooth, proper stack X , denote by ik : Fk ,! X the irreducible
components of the xed locus for this action and by NFk their normal bundles.
The natural map:
P
AC (X ) C(~) ! k AC (Fk ) C(~)
7!
11
ik ctop(NFk ) :
is an isomorphism. Pushing forward equivariantly to the class of a point, we
obtain the Atiyah-Bott integration formula:
Z
[X ]
=
XZ
k
ik :
[Fk ] ctop (NFk )
2.1 Our Set-up
Let C act on a 2 dimensional vector space V via:
t (z0 ; z1 ) = (tz0 ; z1 ):
This action descends on P1 , with xed points 0 = (1 : 0) and 1 = (0 : 1). An
equivariant lifting of C to a line bundle L over P1 is uniquely determined by
its weights fL0 ; L1 g over the xed points.
The canonical lifting of C to the tangent bundle of P1 has weights f1; ,1g .
The action on P1 induces an action on the moduli spaces of Admissible Covers to a parametrized P1 simply by postcomposing the cover map with the
automorphism of P1 dened by t .
The xed loci for the induced action on the moduli space consist of admissible
covers such that anything \interesting" (ramication, nodes) happens over 0
and 1 , or on \non-special" twigs that attach to the main P1 at 0 or 1 .
2.2 Restricting Chow Classes to the Fixed Loci
We want to compute the restriction to various xed loci of the top Chern class
of the bundle
E = R1 f (OP OP (,1)):
The top Chern class c2g+d,1 (E ) splits as
c2g+d,1 (E ) = cg (R1 f OP )cg+d,1 (R1 f OP (,1));
so we will analyze the two terms separately.
There is a standard technique to carry out these computations. To avoid an
overwhelmingly cumbersome notation, we choose to show it only in a particular
example, that will be the most important for our purposes.
Let's consider the xed locus Fg g , consisting of covers where the main P1 is
ramied over 0 and 1 and curves of genus g1 and g2 are attached on either
side. A point in this xed locus is represented in gure 4, where we denote by
X the nodal curve, C1 and C2 the irreducible components over 0 and 1 .
1
1
1 2
12
1
1
X
C2
C1
IP 1
...
...
n2
n1
Figure 4: The xed locus Fg ;g .
1
2
The starting point in analyzing the restriction of the bundle E to this xed
locus is the classical normalization sequence :
0 ! OX ! OC OP OC ! Cn Cn ! 0:
1
1
1
2
2
1) cg (R1 f OP ) :
1
It suces to analyze the long exact sequence in cohomology associated to the
normalization sequence:
0 ! h0 (OX ) ! h0 (OC ) h0 (OP ) h0 (OC ) ! Cn Cn !
! h1 (OX ) ! h1 (OC ) h1 (OC ) ! 0:
Assume that OP is linearized with weights f; g . Then
cg (R1 f OP ) = (,)g g (,)g (,);
where the following notational convention holds:
X
g (n) = (n~)i g,i :
The reason for switching from to , is that h1 (O) are the bers of the dual
bundle to the Hodge bundle, hence the odd degree Chern classes will have a
negative sign.
1
1
1
2
1
2
2
1
1
1
2
2) cg+d,1 (R1f OP (,1)) :
1
In this case we rst want to tensor the normalization sequence by f OP (,1),
and then proceed to analyze the long exact sequence in cohomology:
0 ! h0 (OC ) h0 (OC ) ! Cn Cn !
! h1 (f OP (,1)) ! h1 (OC ) h1 (OP (,d)) h1 (OC ) ! 0:
1
1
1
1
2
1
1
13
2
2
Now, having linearized OP (,1) with weights f; + 1g ,
1
cg+d,1
(R 1 f O
P
g
1 (,1)) = (,) g1 (, )g2
dY
,1 i
d
,
1
(, , 1)~
+ :
1
d
The last term in our contribution, coming from h1 (OP (,d)), is explained in
the following way. Consider a degree d map from P1 to P1 . The target curve
is given the natural C action, and the tautological bundle is linearized with
weights f; + 1g . Now let x and z be local coordinates around 0 for, respectively, the target and the source curve. The expression of the map in local
coordinates is
x = zd:
We see then that z must have weight ,1=d . The vector space h1 (OP (,d))
is (d , 1) dimensional and generated, in local coordinates, by the sections
f1=z; 1=z2 ; :::; 1=zd,1 g . The line bundle over moduli with these bers is trivial,
because P1 is rigid, but it is linearized with weights + i=d ; coming from the
weight of the trivialization of the pullback of OP (,1) in the chart over 0, i=d
from the section 1=z i . Notice that if you were to reproduce this computation
using a local cohordinate over 1 instead, the corresponding weights would now
be ( + 1) , (d , i)=d , which are exactly the same.
1
1
1
2.3 The Euler class of the Normal Bundle to the Fixed Loci
The standard way to carry out this computation is to analyze the deformation
long exact sequence, and identify the ber of the normal bundle to a xed locus
at a particular moduli point to the moving part (the part where the C action
doesn't lift trivially) of the tangent space to the moduli space (corresponding
to the space of rst order deformations of the admissible cover in question).
It's shown in [1], pag 3561, that the deformation theory of admissible covers
corresponds exactly to the deformation theory of the base, genus 0, twisted
curve. The reason for this is that admissible covers are etale covers (in fact
principal Sd -bundles) of the base twisted curve.
Deformations of a genus 0 nodal twisted curve are described as follows: rst
of all, we can deal with one node at a time. For one given node, there are two
dierent potential contributions:
the contribution from moving the node on the main P1 . Doing this innitesimally means moving along the tangent space to the attaching point
on the main P1 . Again, the bundle with ber the tangent space over a
14
given point of P1 is a trivial bundle, but in equivariant cohomology it can
have a purely equivariant rst Chern class, according to the linearization
of the bers. In our particular case, the tangent bundle has weight 1 over
0 and ,1 over 1 , thus producing a contribution of ~ for moving a node
around 0, of ,~ for moving a node around 1 ;
the contribution from smoothing the node. It corresponds to the rst Chern
class of the tensor product of the tangent spaces at the attaching points of the
two curves. Again, we get a ~ contribution from the point on the main P1 ;
the other attaching point x , on the other hand, contributes, by denition, a
, x class.
3 Degree 2
We now carry out the explicit computation of the integral
I2 (g) =
Z
ev1 (1) \ c2g+1 (R1 f (OP OP (,1)));
Admg!2 P1 ;(t
C
1
1 ;t2 ;
1
;t2g+2 )
for all genera, and express the result in generating function form:
1 I (g) X
2
I2(x) =
x2g+1 :
2
g
+
1!
g=0
3.1 The strategy
It is important to notice that, while the nal result is independent of the choice
of the lifting of the C action to the vector bundle E = R1 f (OP OP (,1)),
the intermediate calculations are not. This is in fact the heart of our strategy.
We choose two dierent specic linearizations with the twofold objective of:
limiting a priori the number and the combinatorial complexity of the
contributing xed loci;
obtaining, by equating the calculations with the two linearizations, a
recursive formula for genus g integrals in term of lower genus data.
1
1
3.2 The Localization Set-up
We induce dierent linearizations on the bundle E by choosing dierent liftings
of the C action on the bundles OP and OP (,1). Recall that a linearization
1
1
15
C1
IP 1
...
g
genus 0
twig
IP 1
Figure 5: The xed locus Fg;
of a line bundle over
sentations.
P1
is determined by the weights of the xed bers repre-
Linearization A: We choose to linearize the two bundles as indicated in the
following table:
weight : over 0 over 1
OP (,1) -1
0
OP
0
0
There is only one xed locus Fg; contributing to the localization integral, consisting in a cover of P1 fully ramied over 0 and 1 , and a genus g curve
mapping with degree 2 to an unparametrized P1 sprouting from the point 0.
Figure 5 illustrates the xed locus, and the conventional graph notation to
indicate it.
1
1
The reason for this dramatic collapsing of the contributing xed loci lies in
some standard localization facts:
the ramication condition required over 1 implies that there can be only
one connected component in the preimage of 1 . This translates to the
fact that the localization graph can have at most 1 vertex over 1 ;
the weight 0 linearization of OP (,1) over 1 implies that the localization
graph must have valence 1 over 1 .
nally, let's observe that both bundles have weight 0 over 1 ; the restriction of our bundle to xed loci that have contracted components over 1
1
16
involves the class 2g1 , that vanishes for g > 0 by a famous result by
Mumford ( [11]). The only option is then to have genus 0 over innity.
But a genus zero curve with only two special points is instable, and hence
must be contracted.
Linearization B: We choose to linearize the bundles with weights:
weight : over 0 over 1
OP (,1) -1
0
OP
1
1
1
1
In this case the analysis of the possibly contributing xed loci is similar, except
we can't appeal to Mumford's relation any more. Hence our xed loci will
consist of a copy of P1 ramied over 0 and 1 , with two curves of genus g1 , g2
attached on either side. (And, of course, g1 + g2 = g ). These are the loci Fg ;g
described in gure 4.
1
2
3.3 Explicit evaluation of the Integral and Recursion:
LINEARIZATION A:
Let us rst of all observe that Fg; is naturally isomorphic to
Admg!0;(t ;t ; ;t g ) . Using the computations in section 2, and the standard
equivariant cohomology fact that ev1 (1) = ,~ , we obtain the explicit evalua2
1
2
2 +2
tion of our integral on this xed locus:
I A (g ) =
2
, 12
Z
g g (1)(,~=2) =
Admg!2 0;(t ;t ~;(~;t ,
1 2
Z
)
2g +2 )
g g,1 + g g,2 + + g g,1 :
Admg!2 0;(t
1 ;t2 ;
;t2g+2 )
Just as a convenient notation, let's denote the last integral by L2 (g), so that
I2A (g) = , 21 L2 (g):
(3)
LINEARIZATION B:
17
In this case we have g + 1 dierent types of xed loci, corresponding to all
possible ways of choosing an ordered pair of nonnegative integers adding to g .
We will study separately three situations:
F;g ) This xed locus is naturally isomorphic to 2g + 1 disjoint copies of
Admg!0;(t ;t ; ;t g ) . The evaluation of the integral reads:
Z
g g (,1)(,~=2) =
(,~ , )
F;g
Z
= , 2g 2+ 1
g g,1 + g g,2 + + g g,1 = , 2g 2+ 1 L2 (g):
Adm
2
1
2
2 +2
2
g!
0;(t1 ;t2 ; ;t2g +2 )
Fg ;g , g1 ; g2 6= 0) After keeping track of the combinatorics of the gluing and
1
2
of the possible distributions of the marks, the integral evaluates:
Z
g (,1)g (1)g g (,1)(,~=2) =
~(~ , )(,~ , )
Fg ;g
Z
Z
2
g
+
1
g
2
g
,
1
= , 2g
(, )
g g ,1 +g g ,2 + +g g ,1 =
2
Adm
Adm
1
1
1
2
2
2
1
2
g1 !
0
1
2
2
g2 !
0
2
2
2
2
2
2g + 1 P (g )L (g ):
2 1 2 2
2 2g2
To make the notation a little lighter we omitted the marked points (that
are still there though). Also we choose to denote with P2 the integral of
t to the top power.
Fg;) This is the same xed locus encountered in the computations with linearization A. However, the contribution in this case will be quite dierent:
Z
g (1)g (,1)(,~=2) =
(~ , )
Fg;
Z
(,)g 2g,1 = (,)g+1 12 P2 (g):
= , 12
Adm
:= (,)g1 +1 1
2
g!
0;(t1 ;t2 ; ;t2g +2 )
So allotgether, the integral computed with linearization B is:
g,1
X
1
2
g
+
1
B
g
,
i
I2 (g) = , 2 (2g + 1)L2 (g) , (,)
P2 (g , i)L(i);
2
i
i=0
where we have incorporated the last contribution in the summation by dening
L(0) = 1=2.
18
Lemma 2 For any i , P (i) = 21 .
This follows easily from the fact that the classes that we are using
are pulled back on the space of admissible covers from M0;2g+2 via an etale,
degree 1=2 map (this accounts for the hyperelliptic involution upstairs). The
projective coarse moduli space of M0;2g+2 is M 0;2g+2 , and the two spaces are
birational. It is a classical result that the integral of to the top power on
M 0;2g+2 is one, hence the lemma.
Proof:
We can now equate the results obtained with the two dierent linearizations,
to obtain a recursive formula for the L2 (g)'s.
g,1
X
1
1
, 2 L2 (g) = , 2 (2g + 1)L2 (g) , (,)g,i 2g2+i 1 P2(g , i)L2 (i)
i=0
After a tiny bit of elementary arithmetic we obtain:
L2 (g) = 21g
Pg,1
i=0 (,)
g,i+1 ,2g+1L2 (i):
2i
(4)
3.4 The generating function
We now want to use relation (4) to compute the generating function:
1 L (i) X
2
L2(x) =
x2i+1 :
2
i
+
1!
i=0
Let us rst of all dierentiate this function,
1 L (i) d L (x) = X
2
x2i:
dx 2
2
i
!
i=0
Now let us compute:
g
1
d L (x) sin (x) = X
L2 (i)
2g+1 X(,)g,i+1
x
2
dx
2i!(2g , 2i + 1)! =
g=0
i=0
=
1
X
g=0
x2g+1
1
X
(,)g,i+1 2g2+i 1 2Lg 2+(i)1! = x2g+1 2Lg2+(g1!) = L2 (x):
g=0
i=0
g
X
19
Hence relation (4) translates to the following ODE on the generating function
L2(x):
L02(x) sin (x) = L2(x);
(5)
L2(0) = 0:
This equation integrates to give us L2 (x) = tan(x=2). Finally, recalling (3) we
can conclude:
I2(x) = , 21 tan
,x
2
:
(6)
3.5 A Corollary
Using result (6) it's now easy to compute the generating function for the second
class of integrals we are interested in. Consider
J2 (g) =
Z
c2g+2 (R1 f (OP (,1) OP (,1)));
Admg!2 P1 ;(t
C
1
1
;t2g+2 )
1 ;t2 ;
and the corresponding generating function:
J2(x) =
1 X
g=0
J2 (g) x2g+2 :
2g + 2!
Again, there is a particularly favorable choice of linearizations:
weight : over 0 over 1
OP (,1) -1
0
OP (,1)
0
1
1
1
The only contributing xed loci must have valence 1 both over 0 and 1 .
These are precisely the loci Fg ;g studied above. The explicit computation of
the integral is:
1
2
20
R
, )
: (2g + 2) g ~(g~(1), ( )()(
,~)
g
g1
~
2
g2
,
R
: 2 22gg+2
+1
~
2
g1 g1 (1)( ~2 ) R g2 g2 (,1)(, ~2 )
, )
,~(,~, )
~(~
1
=
1
4 (2g + 2)L2 (g )
=
1 , 2g+2 2 2g1 +1 L2 (g1 )L2 (g2 )
R
, )
: (2g + 2) g ~(g,(,~1), ( )()(
= 41 (2g + 2)L2 (g)
,~)
All previous integrals are computed over the appropriate unparmetrized admissible cover spaces. Adding everything together we obtain the relation:
g
~
2
~
2
g
X
2g + 2 L (i)L (g , i):
1
J (g) = 2
2
2
0 2i + 1
(7)
(Recalling that we have dened L2 (0) = 1=2.)
This relation allows us to obtain the generating function J2 (x). For this purpose
it suces to notice:
J2(x) = 2I2(x)2 = 12 tan2
,x
2
:
(8)
4 Degree 3
In this section we will compute the integral
I3(g) :=
Z
ev(3) (1) \ c2g+2 (R f
Admg!3 P1 ;((3);t ; ;t )
1
2g +2
C
1
(O
P1
OP (,1)));
1
for all genera g , and present the result in generating function form:
I3(x) :=
1
X
I3(g) x2g+2 :
g=0 2g + 2!
21
4.1 The strategy
We will use localization to compute our integral. First of all, we choose an extremely convenient choice of linearizations on the P1 -bundles OP and OP (,1).
This will express our integral in terms of a Hodge integral over only one boundary component of the moduli space.
We then will introduce an auxiliary integral, that we know to vanish for elementary dimension considerations. Evaluating this integral via localization will
produce relations between the integrals I3 (g), for dierent genera g , integrals
in degree 2 and simple Hurwitz numbers.
We are able to transform these relations into a linear dierential equation for
the generating function I3 (x). Finally, solving the ODE with the appropriate
boundary conditions gives us the result.
1
1
4.2 The Localization Set-up
We choose to linearize the bundle as in linearization A in the previous section:
weight : over 0 over 1
OP (,1) -1
0
OP
0
0
1
1
For completely analogous reasons to the degree two case (pag.16), there is only
one xed locus, Fg; , contributing to the localization integral, consisting in a
cover of P1 fully ramied over 0 and 1 , and a genus g curve mapping with
degree 3 to an unparametrized P1 sprouting from the point 0.
The integral then becomes:
Z
g g (1) 29 ~2 2 Z
I3 (g) =
g g,1 3 + g g,2 32 + + g 3g :
=9
~
(
~
,
)
3
Fg;
Adm
3
g!
0;((3);t1 ; ;t2g +2 )
With the sole purpose of keeping track of coecients in a more natural way in
what follows, we give a name to the rightmost integral without the 2=9 in front
of it:
Z
L3 (g) :=
g g,1 3 + g g,2 32 + + g 3g :
Admg!3 0;((3);t ; ;t )
1
2g +2
22
4.3 The auxiliary integral
Let us now consider the following equivariant integral:
Z
ev1 (1) \ c2g+2 (R1 f (OP OP (,1))):
1
Admg!3 P1 ;(t ; ;t )
2g +4
C 1
1
This integral must vanish for dimension reasons. Let us now evaluate this
integral via localization.
We now choose dierent linearizations for the two bundles, as indicated in the
following table.
weight : over 0 over 1
OP (,1) -1
0
OP
1
1
With this choice of linearizations, the explicit evaluation of the integral follows.
We will again be invoking a famous relation by Mumford ( [11]):
g (,1)g (1) = (,)g ~2g :
1
1
g
Fg;0)
0
3
3
3 22gg +
+2
:
Z
Z
(,)g ~2g
~(~ , 3 )
~, 3
Adm0!3 0,;((3)
;t ;t )
Admg!3 0;((3);t ; ;t )
1
2g +2
1 2
2g + 3 1 Z
3 2g + 2 ~ Adm
g! ;
= (,)g+1 2
3
1
Z
2g
0 ((3);t1 ;
2 ~2 =
9
1=
;t2g+2 ) Adm0!3 0;((3);t1 ;t2 )
,
1
= (,)g+1 32 22gg+3
+2 P3;(3) (g )L3 (0) ~ :
g1
Fg ;g )
1
g2
3
2
Z
3 22gg ++32
1
:
(,)g ~2g
~(~ , 3 )
1
1
Admg !3 0;((3);t ; ;t
1
1
2g1 +2 )
23
g g (,1) 2 ~2 =
Z
2
2
,~ ,
;t ; ;t
Admg !3 0;((3)
2
1
2g2 +2 )
3
9
Z
= (,)g +1 23 22gg ++32 ~1
1
Admg ! ;
1
3
Z
2g1
1
;t2g1 +2 )
0 ((3);t1 ;
g g ,1 3 + +g
2
2
2
Admg !3 0;((3);t ; ;t
2
1
2g2 +2 )
,
1
= (,)g +1 23 22gg+3
+2 P3;(3) (g1 )L3 (g2 ) ~ :
1
1
F0;g )
g
0
:
3
Z
3 2g 2+ 3
(~ , 3 )
Adm0!3 0~;((3)
;t ;t )
~, 3
Admg!3 0;((3);t ;,
1 ;t2g +2 )
1 2
Z
2g + 3 1 Z
3 2 ~ Adm 1
!;
= (,)g+1 2
0
g g (,1) 2 ~2 =
Z
1
3
g g,1 3 + + g
Admg!3 0;((3);t ; ;t )
0 ((3);t1 ;t2 )
1
2g +2
,
= , 32 2g2+3 P3;(3) (0)L3 (g) ~1 :
Fg;;)
g
2
1
:
= (,)g 1 1
Z
2g+1
t
2 ~ Adm
g! ; t ; ;t g
Fg ;g ;x)
2
g1
(,)g ~2g 1 ~2 =
~(~ , t ) 2
Admg!3 0;(t ; ;t )
1
2g +4
3
1
Z
2g + 3
2g + 3
0( 1
2 +4 )
g2
2
1
:
24
=
9
(,)g 12 P3;(t) (g) ~1 :
g
3
=
g2
3
=
2 22gg ++33
1
= (,)g1
Z
(,)g ~2g Z
~(~ , t ) Adm
1
1
Admg !3 0;(t ; ;t
1
1
2g2 +4 )
g g (,1) 1 ~2 =
2
2
,~ ,
g ! ; t ; ;t
2
2
0( 1
2g2 +2 )
t
2
Z
2g + 3 1 Z
2g +1
g g ,1 + + g tg,1 =
t
2g1 + 3 ~ Adm
Admg ! ; t ; ;t
g ! ; t ; ;t g
g
1
1
=
F0;g;)
3
0( 1
2
2 2
,
+4 )
2
2
0( 1
2
2 2
2
+2 )
1
(,)g 22gg+3
+3 P3;(t) (g1 )L2 (g2 ) ~ :
1
1
g
2
0
1
:
2 2g 3+ 3
Z
g g (,1) 1 ~2 =
Z
1
,;t )t ) Admg!2 0;(t ;,~;t , )t
Adm0!3 ~0;((t~;
1
4
1
2g +2
2
Z
Z
2
g
+
3
1
=
g g,1 + + g tg,1 =
3 ~ Adm t
! ; t ; ;t Admg! ; t ; ;t g
0
=
3
0( 1
2
4)
0( 1
2 +2 )
,2g+3
3
P3;(t) (0)L2 (g) ~1 :
Finally, adding everything up, we obtain the following relation:
, +3
,
P
P
g,i
(,)g,i+1 P3;(3) (g , i)L3 (i) + gi=0 2g2+3
0 = 23 gi=0 22gi+1
i (,) P3;(t) (g , i)L2 (i): (9)
4.4 The generating function
Now for the less deep but more delicate part of our computation: we need to
extract from relation (9) a dierential equation involving our desired generating
function.
Let's start with a preliminary lemma:
25
Lemma 3 For all g 0:
(1) P3;(3) (g) = 32g :
(2) P3;(t) (g) = (32g+2 , 1)=2:
Proof:
1: Consider the map
Admg!0;((3);t ; ;t g
#
M 0;2g+3 :
3
It's a classical result that
1
Z
M 0;2g+3
2g
1
2 +2
)
= 1:
Since our psi class is just the pull-back of 1 on M0;2g+3 , and this space is
birational to its projective coarse moduli space M 0;2g+3 , our lemma is proven
if we show that has degree 32g . This is a classic Hurwitz number, counting
the number of degree 3 covers of the Riemann sphere with a triple ramication
point and simple ramication otherwise.
The problem is purely combinatorial. We are free to choose a three-cycle in
S3 giving the monodromy of the triple point. The triple point automatically
guarantees that our cover is connected. Then we are free to choose cycles for the
rst (2g +1) simple ramication points. The monodromy of the last ramication
point is determined by the fact that the product of all monodromies should be
the identity. So alltogether we had a choice of 2 32g+1 elements of S3 . We
now need to divide by the conjugation action of S3 on itself, that geometrically
amounts to simply relabelling the sheets of the cover.
Finally we obtain the desired 32g non isomorphic covers.
2: Similarly, we need to count the number of degree 3 covers of P1 with 2g + 4
simple ramication points. Paralleling the previous argument, we can choose
(2g +3) cycles freely. But we have to beware of disconnected covers. These can
happen only if we chose always the same cycle. So in total we have 32g+3 , 3
choices. Dividing now by 6 we obtain our claim.
Let us now translate relation (9) in the language of generating functions. Dene:
L (g) 2g+2 :
L3(x) := P1
g=0 2g+2! x
L (g) 2g+1 :
L2(x) := P1
g=0 2g+1! x
3
2
26
g P ; (g) 2g+2 :
P3;(3) (x) := P1
g=0 (,) 2g+2! x
g P ; t (g) 2g+3 :
P3;(t) (x) := P1
g=0 (,) 2g+3! x
3 (3)
3( )
Then our relation (9) becomes an ordinary dierential equation on the generating functions:
2P
0
0
3 3;(3) L3 , P3;(t) L2 = 0:
(10)
By lemma 3 we can explicitely describe the Hurwitz numbers' generating functions:
x) ;
P3;(3) (x) = 1 , cos(3
9
P3;(t) (x) = 3 sin(x) ,6 sin(3x) :
Also, we do know the generating function for the degree 2 theory, hence:
d tan x =
1 , :
L02(x) = dx
2
2 cos2 x2
Finally, we have reduced our problem to integrating the following:
sin(x),sin(3x) ;
L~03 (x) = 98 (1,3 cos(3
x)) cos ( x )
2
(11)
2
L~3(0) = 0:
This ODE integrates to
!
L3 (x) = 29 4 cos2 ,1x , 1 , 13 :
2
Now let us remember that the generating function I3 (x) is smply (2=9)L3 (x).
After just a little bit of trigonometry clean-up we obtain:
( )
I3(x) = 34 sin
sin ( x )
3
27
x
2
3
2
(12)
4.5 A Corollary
In a completely similar fashion to degree 2, it is possible to obtain from the
previous computation the generating function for the integrals:
J3 (g) =
The answer is:
Z
1
c2g+4 (R f
Admg!3 P1 ;(t ; ;t )
J3(x) = P1
g=0
C
1
J3 (g)
2g+4!
(O 1 (,1) O 1 (,1))):
P
P
2g +4
sin ( )
x2g+4 = 3I (x)2 = 163 sin
( x)
6
2
References
x
2
3
2
(13)
[1] Dan Abramovich, Alessio Corti, Angelo Vistoli, Twisted Bundles and
Admissible Covers, Comm in Algebra 31(8) (2001) 3547{3618
[2] Michael Atiyah, Raoul Bott, The moment map and equivariant cohomology,
Topology 23(1) (1984) 1{28
[3] Jim Bryan, Rahul Pandharipande, The local Gromov-Witten theory of
curves (2004), preprint: math.AG/0411037
[4] Torsten Ekedahl, Sergei Lando, Michael Shapiro, Alek Vainshtein, On
Hurwitz Numbers and Hodge Integrals, C.R. Acad.Sci.Paris Ser.I Math. 328
(1999) 1175{1180
[5] Torsten Ekedahl, Sergei Lando, Michael Shapiro, Alek Vainshtein,
Hurwitz Numbers and intersections on muduli spaces of curves, Invent. Math.
146 (2001) 297{327
[6] Carel Faber, Rahul Pandharipande, Hodge integrals and Gromov-Witten
theory, Invent. Math. 139(1) (2000) 173{199
[7] I Goulden, D M Jackson, Ravi Vakil, Towards the geometry of double Hurwitz numbers (2003), preprint: math.AG/0309440v1
[8] Tom Graber, Ravi Vakil, Hodge integrals, Hurwitz numbers, and virtual localization, Compositio Math. 135 (2003) 25{36
[9] Joe Harris, David Mumford, On the Kodaira dimension of the moduli space
of curves, Invent. Math. 67 (1982) 23{88
[10] Eleny Ionel, Topological recursive relations in H 2 (M ), Invent. Math. 148
(2002) 627{658
[11] David Mumford, Toward an enumerative geometry of the moduli space of
curves, Arithmetic and Geometry II(36) (1983) 271{326
g
28
g;n
Department of Mathematics
University of Utah
155 South 1400 East, Room 233
Salt Lake City, UT 84112-0090
Email: renzo@math.utah.edu
29